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8011A Semester 1 2012 Page 12 of 25

Extended Answer Section

Answer these questions in the answer book(s) provided.Ask for extra books if you need them.

1. (a) Write down a formula for a sinusoidal function f with mean value 2, amplitude 4,period /3 such that f(1) = 2. (4 marks)

(b) Let f(x) = x3 3x2 24x + 7 for x contained in the interval 5 x 5.

(i) Without doing any calculations, briefly justify why f(x) must possess a globalmaximum and a global minimum on the interval 5 x 5. (2 marks)

(ii) Find the global maximum and minimum values off(x) on theinterval 5 x 5.

(4 marks)

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8011A Semester 1 2012 Page 13 of 25

(c) (i) Find the equation of the straight line that passes through the points (2,1)

and (3, 5). (2 marks)

(ii) A researcher has collected data showing how the output y of an experimentdepends upon the input x. After applying a semi-log transformation, thetransformed data fits a linear model and the line of best fit passes through

the points (2,1) and (3, 5). Find y as a function ofx and state whether therelationship you find is a power law or an exponential law.

(3 marks)

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8011A Semester 1 2012 Page 14 of 25

(d) Evaluaten

Xk=0

ar2k, where a = 3 and r = 2. (3 marks)

(e) Evaluate

50Xk=3

1k

1

k 1

. (2 marks)

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8011A Semester 1 2012 Page 15 of 25

2. (a) Suppose that a particle moves in a horizontal line so that its velocity v, measured

in metres per second, as a function of time t is given by:v = t3 3t2 + 3t.

(i) Find all times t where v = 0. (3 marks)

(ii) By doing an integral, find how far the particle travels in the first 5 seconds.(3 marks)

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8011A Semester 1 2012 Page 16 of 25

(iii) Let a =dv

dt

be the acceleration of the particle. Find all times t when the

acceleration a = 0. (2 marks)

(iv) Does the particle ever slow down? That is, is a ever negative? Justify youranswer. (2 marks)

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8011A Semester 1 2012 Page 17 of 25

(b) Define what it means for (a, b) to be a local maximum of the function of two variables

f(x, y). (2 marks)

(c) Let z= f(x, y) = y3 + 12kx2 + 12xy where k is a constant (k 6= 0).

(i) Show that the critical points off(x, y) are (0, 0) and1k2

,2

k

. (4 marks)

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8011A Semester 1 2012 Page 18 of 25

(ii) What kind of critical point is (0, 0)? Justify your answer. (2 marks)

(iii) For what values ofk is1

k2,

2

ka local maximum? Justify your answer.

(2 marks)

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8011A Semester 1 2012 Page 19 of 25

(d) Using thatd

dxe

g(x)

= g0(x)eg(x), prove that

d

dx

(2x) = (ln2)2x. (2 marks)

(e) Given f(x, y) = e

y

2

sin(xy2)

, find fx(0,

) and fy(0,

). (3 marks)

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8011A Semester 1 2012 Page 20 of 25

3. (a) Let f(x, y) be a function of two variables with the property that

f(a, b) = fx(a, b)a + fy(a, b)b,for all (a, b) 2 R2.

(i) Suppose fx(1, 4) = 2 and fy(1, 4) = 3. Evaluate f(1, 4) and hence writedown the equation of the tangent plane at (1, 4, f(1, 4)). (4 marks)

(ii) Show that the tangent plane to any point on the surface z= f(x, y) necessarilypasses through the origin (0, 0, 0) in R3. (3 marks)

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8011A Semester 1 2012 Page 21 of 25

(b) Evaluate the following improper integral, if possible: Z0

1

3x2ex3

dx. (4 marks)

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8011A Semester 1 2012 Page 22 of 25

(c) (i) Solve the equation cos = sin 2 for 0 2[Hint: sin2 = 2 sin cos .] (4 marks)

(ii) Using the result of part (i) or otherwise, sketch the region bounded by the

curves y = cos 2x and y = sin 4x, and the straight lines x = 0 and x = 4

.

(5 marks)

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8011A Semester 1 2012 Page 23 of 25

(iii) Find the area of the region described in part (ii). (6 marks)

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8011A Semester 1 2012 Page 24 of 25

This blank page may be used if you need more space for your an-

swers.

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8011A Semester 1 2012 Page 25 of 25

This blank page may be used if you need more space for your an-

swers.

End of Extended Answer Section

This is the last page of the question paper.

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8011B Semester 1 2012 Multiple Choice Answer Sheet

0 0

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

Write your

SID here !

Code your

SID into

the columns

below each

digit, by

filling in the

appropriate

oval.

Answers !

a b c d e a b c d e

Q1

Q2

Q3

Q4

Q5

Q6

Q7

Q8

Q9

Q10

Q11

Q12

Q13

Q14

Q15

Q16

Q17

Q18

Q19

Q20

Q21

Q22

Q23

Q24

Q25

The University of SydneySchool of Mathematics and

Statistics

MATH1011 Applications of

Calculus

Family Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Seat Number: . . . . . . . . . . . . . . . . .

Indicate your answer to each question byfilling in the appropriate oval.

This is the first and last page of this answer sheet

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Correct Responses to MC Component of

MATH1011 Applications of Calculus8011: Semester 1 2012

Q1 ! b

Q2 ! e

Q3 ! c

Q4 ! e

Q5 ! d

Q6 ! a

Q7 ! e

Q8 ! d

Q9 ! e

Q10 ! a

Q11 ! b

Q12 ! c

Q13 ! d

Q14 ! d

Q15 ! c

Q16 ! b

Q17 ! e

Q18 ! a

Q19 ! b

Q20 ! a

Q21 ! c

Q22 ! a

Q23 ! d

Q24 ! b

Q25 ! e